1. ISS of Switched Systems and Application to Adaptive Control
Linh Vu, Debasish Chatterjee, Daniel Liberzon
Coordinated Science Laboratory,
University of Illinois, Urbana-Champaign, U.S.A
CDC 2005

2. Switched Nonlinear Systems
where
is a switching signal
and is some index set
where
ISS:
-ISS:
-iISS:
LTI systems: Hespanha-Morse. “Stability of switched systems with average dwell-time”. (‘99)

3. Slow switching Average dwell-time switching where
is the number of switches in
is a chatter bound
is the average dwell-time
Hespanha-Morse. “Stability of switched systems with average dwell-time”. (‘99)

4. ü ý þ Main result Theorem 1: Suppose that there exist such that Û ISS
ISS if
Then the switched system is
-ISS if
-iISS if
ISS ® AS : Hespanha-Morse. (‘99)

5. Application: Supervisory Control
A parameterized plant with unknown parameters
Goal: stabilize the plant when and guarantee bounded when is bounded.
Control Scheme:
For each design a stabilizing controller
Design a multi-estimator
Estimation error

6. Controllers + Multi-estimator = Injected systems
fixed
Plant P
Controller
Multi-
estimator C E
Controllers + Multi-estimator = Injected systems
since

7. ü ý þ + + A1: (criterion for the multi-estimator design)
A2: The injected systems satisfy ISS hypotheses of Theorem 1 with respect to
(criteria for controllers + multi-estimator design) 2 1
If we have some switching mechanism that provides
smallness of w.r.t ü ý þ Þ
bounded
Thm 1 + A2 1
ISS type properties of the switched injected system with respect to
then 2 + A1
smallness of

8. ü ý þ A3: The plant state is bounded when are bounded. A3Þ +
Scaled-independent hysteresis switching logic gives:
Hespanha-Morse. (‘99) 1
for some chosen 2
(smallness of w.r.t )

9. Theorem 2: Suppose that the plant satisfies A3
the controllers and the multi-estimator satisfy A1 and A2.
Then under the scaled-independent hysteresis switching logic, all continuous states of the closed-loop systems are bounded for arbitrary initial conditions and bounded disturbance.
Further, if then can make

10. Example:
Controllers:
Multi-estimator:

11. Conclusion
ISS type properties of switched system under average dwell-time switching
Application to supervisory control of nonlinear uncertain systems in the presence of disturbances
Future work:
Relax the requirement of the existence of a constant for nonlinear systems
Include unmodeled dynamics (non-exact matching)
Use other slow switching mechanisms